Evaluate the limit of sums by first expressing it as a definite integral. Evalua

Question

Evaluate the limit of sums by first expressing it as a definite integral. Evaluate the limit of sums by first expressing it as a definite integral. Recall the definition of a Riemann Sum for the function f(x) on the interval [a, b]. A Riemann Sum is a series of the form , where {x0, x1, xn} is a partition of [a, b], , and t; is an element of the subinterval . In the given sum, what are f(t), t;, and tor? What interval does the set partition? What definite integral does this sum converge to, and how can the Fundamental Theorem be used to evaluate it?

Explanation / Answer

the can be written as lim 1/n + 3/n^3 summation (i^2)

=> 1/n +3 n*(n+1)(2n+1)/6n^3
=>1/n +1/2*(1+1/n)(2+1/n)
=> 1

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